Abstract
We survey several Talagrand type inequalities and their application to influences with the tool of hypercontractivity for both discrete and continuous, and product and non-product models. The approach covers similarly by a simple interpolation the framework of geometric influences recently developed by N. Keller, E. Mossel and A. Sen. Geometric Brascamp-Lieb decompositions are also considered in this context.
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C. Ané, S. Blachère, D. Chafaï, P. Fougères, I. Gentil, F. Malrieu, C. Roberto, G. Scheffer, Sur les inégalités de Sobolev logarithmiques. (French. Frech summary) [Logarithmic Sobolev inequalities] Panoramas et Synthèses [Panoramas and Syntheses] vol. 10 (Société Mathématique de France, Paris, 2000)
D. Bakry, in L’hypercontractivité et son utilisation en théorie des semigroupes. Ecole d’Eté de Probabilités de Saint-Flour, Lecture Notes in Math., vol. 1581 (Springer, Berlin, 1994), pp. 1–114
D. Bakry, M. Ledoux, Lévy-Gromov’s isoperimetric inequality for an infinite dimensional diffusion generator. Invent. Math. 123, 259–281 (1996)
D. Bakry, I. Gentil, M. Ledoux, Forthcoming monograph (2012)
F. Barthe, P. Cattiaux, C. Roberto, Interpolated inequalities between exponential and gaussian Orlicz hypercontractivity and isoperimetry. Revista Mat. Iberoamericana 22, 993–1067 (2006)
F. Barthe, D. Cordero-Erausquin, M. Ledoux, B. Maurey, Correlation and Brascamp-Lieb inequalities for Markov semigroups. Int. Math. Res. Not. IMRN 10, 2177–2216 (2011)
W. Beckner, Inequalities in Fourier analysis. Ann. Math. 102, 159–182 (1975)
S. Bobkov, C. Houdré, A converse Gaussian Poincaré-type inequality for convex functions. Statist. Probab. Lett. 44, 281–290 (1999)
A. Bonami, Étude des coefficients de Fourier des fonctions de Lp(G). Ann. Inst. Fourier 20, 335–402 (1971)
P. Diaconis, L. Saloff-Coste, Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Prob. 6, 695–750 (1996)
D. Falik, A. Samorodnitsky, Edge-isoperimetric inequalities and influences. Comb. Probab. Comp. 16, 693–712 (2007)
J. Kahn, G. Kalai, N. Linial, The Influence of Variables on Boolean Functions. Foundations of Computer Science, IEEE Annual Symposium, 29th Annual Symposium on Foundations of Computer Science (FOCS, White Planes, 1988), pp. 68–80
N. Keller, E. Mossel, A. Sen, Geometric influences. Ann. Probab. 40(3), 1135–1166 (2012)
D. Kiss, A note on Talagrand’s variance bound in terms of influences. Preprint (2011)
M. Ledoux, The geometry of Markov diffusion generators. Ann. Fac. Sci. Toulouse IX, 305–366 (2000)
M. Ledoux, in The Concentration of Measure Phenomenon. Math. Surveys and Monographs, vol. 89 (Amer. Math. Soc., Providence, 2001)
T.Y. Lee, H.-T. Yau, Logarithmic Sobolev inequality for some models of random walks. Ann. Probab. 26, 1855–1873 (1998)
E. Milman, On the role of convexity in isoperimetry, spectral gap and concentration. Invent. Math. 177, 1–43 (2009)
E. Milman, Isoperimetric and concentration inequalities – Equivalence under curvature lower bound. Duke Math. J. 154, 207–239 (2010)
R. O’Donnell, K. Wimmer, KKL, Kruskal-Katona, and Monotone Nets. 2009 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2009) (IEEE Computer Soc., Los Alamitos, 2009), pp. 725–734
R. O’Donnell, K. Wimmer, Sharpness of KKL on Schreier graphs. Preprint (2011)
R. Rossignol, Threshold for monotone symmetric properties through a logarithmic Sobolev inequality. Ann. Probab. 34, 1707–1725 (2006)
G. Royer, An initiation to logarithmic Sobolev inequalities. Translated from the 1999 French original by Donald Babbitt. SMF/AMS Texts and Monographs, vol. 14 (American Mathematical Society, Providence, RI (Société Mathématique de France, Paris, 2007)
M. Talagrand, On Russo’s approximate zero-one law. Ann. Probab. 22, 1576–1587 (1994)
Acknowledgements
We thank F. Barthe and P. Cattiaux for their help with the bound (26), and R. Rossignol for pointing out to us the references [20, 21]. We also thank J. van den Berg and D. Kiss for pointing out that the techniques developed here cover the example of the complete graph and for letting us know about [14].
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Cordero-Erausquin, D., Ledoux, M. (2012). Hypercontractive Measures, Talagrand’s Inequality, and Influences. In: Klartag, B., Mendelson, S., Milman, V. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2050. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29849-3_10
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DOI: https://doi.org/10.1007/978-3-642-29849-3_10
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